Rank error-correcting code - Revision history

95.108.239.101 at 15:36, 2 December 2014

2014-12-02T15:36:23Z

5.146.76.10: /* Applications */

2014-01-03T13:24:20Z

Applications

← Older revision		Revision as of 15:24, 3 January 2014
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			In mathematics, the '''Grothendieck–Ogg–Shafarevich formula''' describes the [[Euler characteristic]] of complete curve with coefficients in an [[abelian variety]] or [[constructible sheaf]], in terms of local data involving the [[Swan conductor]]. {{harvs\|txt\|last=Ogg\|author-link=Andrew Ogg\|year=1962}} and {{harvs\|txt\|last=Shafarevich\|authorlink=Igor Shafarevich\|year=1961}} proved the formula for abelian varieties with tame ramification over curves, and {{harvs\|txt\|last=Grothendieck\|authorlink=Alexander Grothendieck\|year=1977\|loc=Exp. X formula 7.2}} extended the formula to constructible sheaves over a curve {{harv\|Raynaud\|1965}}.

			==Statement==

			Suppose that ''F'' is a constructible sheaf over a genus ''g'' smooth projective curve ''C'', of rank ''n'' outside a finite set ''X'' of points where it has stalk 0. Then
			:<math>\chi(C,F) = n(2-2g) -\sum_{x\in X}(n+Sw_x(F))</math>
			where ''Sw'' is the Swan conductor at a point.

			==References==

			*{{citation
			\| last = Grothendieck
			\| first = Alexandre
			\| authorlink = Alexandre Grothendieck
			\| title = Séminaire de Géométrie Algébrique du Bois Marie - 1965-66 - Cohomologie l-adique et Fonctions L - (SGA 5) \|series=Lecture notes in mathematics \|volume = 589
			\| year = 1977
			\| publisher = [[Springer Science+Business Media\|Springer-Verlag]]
			\| location = Berlin; New York
			\| language = French
			\| pages = xii+484
			\| isbn = 3540082484
			\| nopp = true
			\|doi = 10.1007/BFb0096802
			}}
			*{{Citation \| last1=Ogg \| first1=A. P. \| title=Cohomology of abelian varieties over function fields \| jstor=1970272 \| mr=0155824 \| year=1962 \| journal=[[Annals of Mathematics\|Annals of Mathematics. Second Series]] \| issn=0003-486X \| volume=76 \| pages=185–212}}
			*{{Citation \| last1=Raynaud \| first1=Michel \| author1-link=Michel Raynaud \| title=Séminaire Bourbaki, Vol. 9 \| url=http://www.numdam.org/item?id=SB_1964-1966__9__129_0 \| publisher=[[Société Mathématique de France]] \| location=Paris \| series=Exp. No. 286 \| mr=1608794 \| year=1965 \| chapter=Caractéristique d'Euler-Poincaré d'un faisceau et cohomologie des variétés abéliennes \| pages= 129–147}}
			*{{Citation \| last1=Shafarevich \| first1=Igor R. \| title=Principal homogeneous spaces defined over a function field \| mr=0162806 \| year=1961 \| journal=Akademiya Nauk SSSR. Trudy Matematicheskogo Instituta imeni V. A. Steklova \| issn=0371-9685 \| volume=64 \| pages=316–346}}

			{{DEFAULTSORT:Grothendieck-Ogg-Shafarevich formula}}
			[[Category:Elliptic curves]]
			[[Category:Abelian varieties]]

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2012-03-22T02:07:29Z

General fixes, removed orphan tag using AWB (8032)

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